Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I'd like to become conversant in a wide variety of serious mathematics, but i'm currently one of those students who did very poorly on mathematical subjects in school, never completing even basic algebra. My failure at math has been an ever-present awareness throughout my entire life, and i'd like to go beyond it. I'm in a technical field and my lack of good math foundations haven't crippled me, but i'd like to fill that hole.

I've started to get really serious about (re)learning mathematics, from my somewhat poor arithmetic foundations. I understand that there are essentially two schools of thought: The school of thought that suggests arithmetic->algebra->geometry/calculus (in either order)->analysis->everything else, and the school of thought that suggests an understanding of Set Theory and Logic (being the foundation of mathematics) is important.

I know that i'll eventually learn set theory and logic, but is there some benefit to learning these things before even trying the rest of mathematics? If one were to learn those things, what's the best place for a complete beginner to start? If I were to choose set theory, would ZFC be the only good option for a complete beginner?

share|improve this question
5  
For a complete beginner, unformalized (but precise) set theory should precede any introduction to ZFC. Indeed, one can do very good work in mathematics without being aware of the existence of ZFC. I would also advise work in other subjects before set theory. some of the subjects will use set-theoretic notation. That's not too hard to absorb while studying something else. –  André Nicolas Apr 11 '13 at 1:49
    
I'd do some geometry as soon as you feel comfortable with algebra. It's the funnest. And you can prove as much or as little as you like. –  user66345 Apr 11 '13 at 1:55
    
This may be helpful. –  EuYu Apr 11 '13 at 2:05
1  
I second @AndréNicolas' suggestion. It will be very hard to self-study set theory without motivation. Perhaps just follow the standard sequence. You can read set theory on the side, but don't worry about too much about forcing yourself to read it if you find your interest waning. –  user66345 Apr 11 '13 at 2:17
    
On the whole I agree with @André’s comment above. If you want to have on the side a book that gives a decent but relatively informal introduction to set theory, logic, and some closely related topics, you could do worse than Bob Stoll’s Set Theory and Logic, available in an inexpensive Dover paperback. –  Brian M. Scott Apr 11 '13 at 3:46
show 1 more comment

3 Answers

A primary goal of "Set Theory and Logic" (I put this in quotations because I get the sense you are referring to a particular school of thought, and not just the pure subjects on their own) is to give foundation and motivation to the structures and systems of numbers that we commonly use. As the most basic example, efforts have been made to define natural numbers in terms of sets. Another basic example is the effort to define Mathematics as an extension of Logic.

While these are highly interesting studies, I would classify these types of studies more under the heading of "meta-mathematics" or foundations of mathematics. In essence, this type of study works backwards from the familiar world of numbers and mathematical areas we know, and attempts to ground these structures in well-defined "fundamental" ideas (sorry I have to be vague here, but this stuff is abstract!).

At any rate, from what I've said, you can get the sense that these types of study are not the typical areas a beginner should engage in, unless that beginner be of a more philosophical disposition; in other words, these areas are of a broader nature, and have a different conceptual "flavor". They seek to unify mathematical structures into more basic structures.

On the other hand, the "typical" mathematician works within established fields of math; that is to say, he uses and manipulates the structures and symbols given to him, in an attempt to discover deeper connections and new relationships. He is not usually concerned with foundations, that is a completely seperate study.

So, after I've said all that, my practical advice is to move along the "Algebra -> Pre-Calculus -> Calculus -> etc." route. That gives one the necessary tools for advanced study, and it familiarizes one (at a nice pace!) with what mathematicians really do. And IMHO, Calculus is absolutely essential in this path, because studying that results in a certain understanding and maturity in math that one will need throughout the rest of his mathematical career (e.g., the notions of limit and derivative in Calculus are really fundamental, and are great examples of mathematical intuition and thought).

Just a side note, I am not implying that the independent fields of Logic and Set Theory, as subjects on their own, are "deeply abstract" in the sense I described above (i.e., relate to the foundations of math); but that being said, I do not think they serve as beginning studies either. I believe they fall under the "etc." in the path I mentioned above.

Hope this helps you figure out how you'd like to proceed! Good Luck.

share|improve this answer
    
Thanks for the notes. My rationale is that understanding set theory and logic may be a better "bridge" for me as in some ways the philosophical concepts may be easier to understand than jumping right in. When I failed my mathematics courses, I was always confused as to what math really was, or "why" certain things were the way that they were; now understanding math as just a constructive method of dealing with problems has made things easier but i'm very weak on applications. My hypothesis is that understanding set theory and logic might solve these unanswered questions for me. –  Arima Apr 11 '13 at 2:17
    
@Arima : I have to warn you though that these foundation-type studies often employ the ideas from the fields of set theory and logic. So you'd need those first. And if you're going to study those, then you need some basic skills in arithmetic, algebra etc...That's why I was implying the foundations-type study of math is for more advanced studiers. But far be it from me to discourage you if its the "philosophy-type" stuff that interests you; I just mean to say its not the typical math. –  Coffee_Table Apr 11 '13 at 2:24
add comment

It depends on what you want to do with mathematics. If your aim is to build a repertoire of problem-solving tools for applications in science and industry, you will probably get all the set theory and logic you will ever need from the introductory chapter of any good algebra or calculus textbook. If reading and writing detailed proofs (e.g. proving there are an infinite number of prime numbers) is important to you, you may need more than that.

I believe it is possible to get an undergraduate degree in pure mathematics without ever once dealing at any length with foundational issues such as the axioms of set theory (ZFC, etc.). It really is quite a specialized interest. I would stick to application-oriented algebra, calculus and statistics for now. A few years of that and ZFC just might make sense to you. But don't worry if it doesn't.

share|improve this answer
    
Where I studied for my B.Sc. and M.Sc. you can even get through a Ph.D. without taking a course on axiomatic set theory, or any foundational topic. Of course this much is not very easy, but it's possible. –  Asaf Karagila Apr 11 '13 at 15:17
add comment

If you would like to learn set theory and logic. You can start from the Kunen's book: Set Theory. It is a good book for a beginner. Hope it be helpful for you.

share|improve this answer
6  
That's a book for a beginning graduate course. The OP said he hasn't completed high school algebra! –  user66345 Apr 11 '13 at 1:57
    
Do you have a better set theory book for him? –  Paul Apr 11 '13 at 1:59
3  
I suggest beginning with naive set theory. Halmos may be more appropriate or so would Enderton for that matter when compared to Kunen's book. –  EuYu Apr 11 '13 at 2:00
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.